A Portal Special Presentation- Geometric Unity: A First Look: Difference between revisions

''[https://youtu.be/Z7rd04KzLcg?t=6404 01:46:44]''<br>

Now, the question is, we've integrated so tightly with the matter fields, we have to ask ourselves the question, "Can we see unification here?"

''[https://youtu.be/Z7rd04KzLcg?t=6421 01:47:01]''<br>

Let's define matter content in the form of Omega-0 tensored in the spinors, which is a fancy way of saying spinors, together with a copy of the one-forms tensored in the spinors. And let me come up with two other copies of the same data—so I'll make \(\Omega^{d-1}\) just by duality, so imagine that there's a Hodge star operator.

''[https://youtu.be/Z7rd04KzLcg?t=6463 01:47:43]''<br>

And, when I was a little kid, I had the Soma cube. I don't know if you've ever played with one of these things, they're fantastic. And I later found out that this guy who invented the Soma cube—which you had to put together as pieces—there was one piece that looked like this object. And he was like this amazing guy in the Resistance during World War II. So I would like to name this, the <strong>somatic complex</strong> after, I guess his name is Piet Hein. So this complex—I'm going to choose to start filling in some operators: the exterior derivative coupled to a connection, but in the case of spinors we're going to put a slash through it. Let's make this the identity.

''[https://youtu.be/Z7rd04KzLcg?t=6506 01:48:26]''<br>

We'd now like to come up with a second operator here. And this second operator here should have the property that the complex should be exact and the obstruction to it being a true complex—to nilpotency—should be exactly the generalization of the Einstein equations, thus unifying the spinorial matter with the intrinsic replacement for the curvature equations.

''[https://youtu.be/Z7rd04KzLcg?t=6539 01:48:59]''<br>

Well, we know that \(\unicode{x2215}\kern-0.5em d_A\) composed with itself is going to be the curvature, and we know that we want that to be hit by a shiab operator. And if shiab is a derivation, you can start to see that that's going to be curvature, so you want something like \(F_A\) followed by shiab over here to cancel. Then you think okay, how am I going to get at getting this augmented torsion? And, then you realize that the information in the inhomogeneous gauge group, you actually have information not for one connection, but for two connections.

''[https://youtu.be/Z7rd04KzLcg?t=6584 01:49:44]''<br>

So in one case, I can do plus star to pick up the \(A_{\pi}\). But I am also going to have a derivative operator if I just do a star operation, so I need another derivative operator to kill it off here. So I'm going to take minus the derivative with respect to the connection, which defines a connection one-form as well as having the same derivative coming from the Levi-Civita connection on \(U\).

''[https://youtu.be/Z7rd04KzLcg?t=6615 01:50:15]''<br>

So in other words, I have two derivative operators here. I have two ad-value one-forms. The difference between them has been to be a zeroth-order, and it's going to be precisely the augmented torsion. And that's the same game I'm going to repeat here.

''[https://youtu.be/Z7rd04KzLcg?t=6652 01:50:52]''<br>

So I'm going to do the same thing here. I'm going to define a bunch of terms where, in the numerator, I'm going to pick up the \(\pi\) as well as the derivative, in the denominator, because I have no derivative here, I'm going to pick up this \(h^{-1} d_{A_0} h\).

''[https://youtu.be/Z7rd04KzLcg?t=6677 01:51:17]''<br>

I'm going to do that again on the other side. There are going to be plus and minus signs, but it's a magic bracket that knows whether or not it should be a plus sign or a minus sign and I apologize for that, but I'm not able to keep that straight. And then there's going to be one extra term, where all these \(T\)s have the \(\epsilon\) and \(\pi\)s.

''[https://youtu.be/Z7rd04KzLcg?t=6710 01:51:50]''<br>

Okay. So some crazy series of differential operators on the northern route... So if you take the high road or you take the low road, when you take the composition of the two, the differential operators fall out and you're left with an obstruction term that looks like the Einstein field equation.

''[https://youtu.be/Z7rd04KzLcg?t=6741 01:52:21]''<br>

Well, that's pretty good, if true. Can you go farther? Well, look at how close to this field content is to the picture from deformation theory that we learned about in low dimensions. The low-dimensional world works by saying that symmetries map to field content map to equations, usually in the curvature.

[[File:GU Presentation Sym-Fld-Eq Diagram.png|thumb|right]]

''[https://youtu.be/Z7rd04KzLcg?t=6774 01:52:54]''<br>

And when you linearize that, if you are in low enough dimension, you have \(\Omega^0\), \(\Omega^1\), sometimes \(\Omega^0\) again, and then something that comes from \(\Omega^2\), and if you can get that sequence to terminate by looking at something like a half-signature theorem or a bent back de Rham complex in the case of dimension three, you have an Atiyah-Singer theory. And remember we need some way to get out of infinite dimensional trouble, right? You have to have someone to call when things go wrong overseas and you have to be able to get your way home. And in some sense, we call on Atiyah and Singer and say we're in some infinite dimensional space, can't you cut out some finite dimensional problem that we can solve even though we start getting ourselves into serious trouble? And so we're going to do the same thing down here. We're going to have \(\Omega^0(ad)\), \(\Omega^1(ad)\), direct sum \(\Omega^0(ad)\), \(\Omega^{d−1}(ad)\), and it's almost the same operators.

''[https://youtu.be/Z7rd04KzLcg?t=6849 01:54:09]''<br>

And this is now not just a great guess, it's actually the information for the deformation theory of the linearized replacement of the Einstein field equations. So this is computing the Zariski tangent space just as if you were doing self-dual theory or Chern-Simons theory. You've got two somatic complexes, right? One of them is Bose, one of them is Fermi. The obstruction to both of them is a common generalization of the Einstein field equations. What's more is if you start to think about this, this is some version of Hodge theory with funky operators. So you can ask yourself, "Well, what are the harmonic forms in a fractional spin context?"

''[https://youtu.be/Z7rd04KzLcg?t=6903 01:55:03]''<br>

Well, they're different depending upon whether you take the degree-zero piece together with the degree-two piece, or you take the degree-one piece. Let's just take the degree-one piece. You get some kind of equation—so, I'm going to decide that I have a \(\zeta\) field, which is an \(\Omega^1(\unicode{x2215}\kern-0.55em S)\), and a field \(\nu\), which always strikes me as a Yiddish field. \(\nu\) is \(\Omega^0(\unicode{x2215}\kern-0.55em S)\).

<div style="text-align: center; margin-left: auto; margin-right: auto;">$$ \zeta \in \Omega^1(\unicode{x2215}\kern-0.55em S) $$</div>

<div style="text-align: center; margin-left: auto; margin-right: auto;">$$ \nu \in \Omega^0(\unicode{x2215}\kern-0.55em S) $$</div>

''[https://youtu.be/Z7rd04KzLcg?t=6947 01:55:47]''<br>

What equation would they solve if we were doing Hodge theory relative to this complex? The equation would look something like this.

<div style="text-align: center; margin-left: auto; margin-right: auto;">$$ *d_A^* \zeta = *\nu $$</div>

''[https://youtu.be/Z7rd04KzLcg?t=6970 01:56:10]''<br>

There'd be one equation that was very simple, and then there'd be one equation that would be like really hard to guess.

''[https://youtu.be/Z7rd04KzLcg?t=7068 01:57:48]''<br>

Now if you have something like that, that would be a hell of a Dirac equation. Right? You've got differential operators over here. You've got differential operators—I guess I didn't write them in—but you would have two differential operators over here, and you'd have this differential operator coming from this Maurer-Cartan form. So I apologize, I'm being a little loose here, but the idea is you have two of these terms are zeroth-order, three of these terms would be first-order, and on this side (the left side), one term would be first-order.

''[https://youtu.be/Z7rd04KzLcg?t=7107 01:58:27]''<br>And that's not there, that was a mistake. Oh no, sorry. That was a mistake, calling it a mistake. These are two separate equations, right? So you have two separate fields, \(\nu\) and \(\zeta\), and you have a coupled set of differential equations that are playing the role of the Dirac theory, coming from the Hodge theory of a complex, whose obstruction to being [a] cohomology theory would be the replacement to the Einstein field equations, which would be rendered gauge invariant on a group relative to a tilted subgroup.

''[https://youtu.be/Z7rd04KzLcg?t=7152 01:59:12]''<br>

So now we've dealt with two of the three sectors. Is there any generalization of the Yang-Mills equation? Well, if we were to take the Einstein field equation generalization and take the norm square of it—oh, there's some point I should make here, just one second. I've been treating this as if everything is first-order, but what really happens here is that you've got symmetries. You've got symmetric field content, you've got ordinary connections, and we're neglecting to draw the fact that there have to be equations here too. These equations are first-order. So why do we get to call this a first-order theory if there are equations here which are of second-order? Well, it's not a pure first-order theory, but when I say a first-order theory in this context, what I really mean is that the second-order equations are completely redundant on the first-order equations by virtue of the symmetry principle. That is, any solution of the first-order equation should automatically imply a solution of the second-order equations. So from that perspective, I can pretend that this isn't here because it is sufficient to solve the first-order equations.

''[https://youtu.be/Z7rd04KzLcg?t=7249 02:00:49]''<br>

So I can now look—let's call that entire replacement, which we previously called \(\alpha\), I'm going to set \(\alpha\) equal to \(\Upsilon\), because I've actually been using \(\Upsilon\), the portion of it that is just the first-order equations, and take the norm squared of that, that gives me a new Lagrangian. And if I solve that new Lagrangian, it leads to equations of motion that look like exactly what we said before.

''[https://youtu.be/Z7rd04KzLcg?t=7288 02:01:28]''<br>

And it ends up defining an operator that looks something like this, \(d_A^∗\) the adjoint of the shiab operator.

''[https://youtu.be/Z7rd04KzLcg?t=7321 02:02:01]''<br>

So in other words, this piece gives you some portion that looks like, right, from the swervature tensor there's going to be some component that's playing the role of Einstein's field equations directly, and the Ricci tensor, but generalized. And then you're going to have some differential operator here, so that the replacement for the Yang-Mills term, instead of \(d^∗_A F_A\), you've got these two shiab and an adjoint shiab together in the center, generalizing the Yang-Mills theory. Then you say well, how come we don't just see the Yang-Mills theory? Why don't we see general relativity as well? But in the full expansion there's also a term that's zeroth-order that's effectively acting like the identity which hits this as well. So you have one piece that looks like the Yang-Mills theory, and in these second-order equations you also have a piece that looks like the Einstein theory. And this is in the vacuum equations. So then the question is how do you see the Dirac theory coming out of this?

''[https://youtu.be/Z7rd04KzLcg?t=7387 02:03:07]''<br>

And so what we're just trying to put together now before we come out with the manuscript for this is, putting these two elliptic complexes together, the Dirac terms go between the two complexes, right? So the idea is that the stress-energy tensor should be the up-and-back term and the Dirac equations should come out of the term that goes up-and-over versus the term that goes over-and-up, and you need some cancellations to make sure that everything is of zeroth-order, properly invariant, etc. And that's taking a little time because frankly, I'm not good at keeping track of indices, minus signs, left-right—it's a learning-disabled nightmare.